Why it is a question
THEORY-313 now states Bach’s sandwich D(p̃‖q̃) ≤ D(Σ_p‖Σ_q) ≤ D(p‖q) for unit-diagonal kernels. It also states the gap bound: D(p‖q) − D(p̃‖q̃) is at most a constant times sup_x ∫h(x, y)d(x, y)² dy. On the torus with Fourier coefficients proportional to e^(−σ‖ω‖₁), that integral is at most dσ²/16. The audit re-derived all three (LIT-255, NOTE-837).
Bach asserts that the rate extends to every translation-invariant kernel on [0, 1]^d and to translation-invariant kernels on compact spaces, and sketches a leverage-score argument. Neither is carried through. The quantity that has to be controlled is ∫h(x, y)d(x, y)² dy, with h the squared normalized kernel, and that depends on how fast the kernel’s spectrum decays.
What would answer it
A bound of ∫h d² = O(σ²) for a stated wider class of translation-invariant kernels would answer it yes for that class: for example, kernels whose Fourier coefficients are a fixed decreasing profile rescaled by σ, or Gaussian kernels on the torus. So would the same bound for invariant kernels on a compact group or homogeneous space, or a later paper that proves either. A translation-invariant unit-diagonal kernel family whose gap shrinks more slowly than σ² as its bandwidth shrinks would answer it no. Numerical agreement between kernel and Shannon estimates on examples cannot settle it, since the inequalities permit agreement at any bandwidth.